[Paper Review] Equitable block colourings

Paola Bonacini, Lucia Marino|arXiv (Cornell University)|Jun 20, 2014

graph theory and CDMA systems5 references3 citations

TL;DR

This paper investigates equitable block colourings in 4-cycle systems of order $v = 1 + 8k$, focusing on $c$-colourings of type $s$ where $s$ divides $k$. It constructs equitable colourings for $c \in \{s, s+1, \dots, \lfloor (2s^2 + s)/3 \rfloor\}$ using structured block decompositions and modular arithmetic, proving that the minimum $s$-chromatic index is $s$ and establishing bounds on the maximum index.

ABSTRACT

Let $Σ=(X,\mathcal B)$ a $4$-cycle system of order $v=1+8k$. A $c$-colouring of type $s$ is a map $ϕ\colon \mathcal B ightarrow \mathcal C$, with $C$ set of colours, such that exactly $c$ colours are used and for every vertex $x$ all the blocks containing $x$ are coloured exactly with $s$ colours. Let $4k=qs+r$, with $q,r\ge 0$. $ϕ$ is \emph{equitable} if for every vertex $x$ the set of the $4k$ blocks containing $x$ is parted in $r$ colour classes of cardinality $q+1$ and $s-r$ colour classes of cardinality $q$. In this paper we study colourings for which $s|k$, giving a description of equitable block colourings for $c\in \{s,s+1,\dots,\lfloor frac{2s^2+s}{3} floor \}$.

Motivation & Objective

• To characterize equitable block colourings in $4$-cycle systems of order $v = 1 + 8k$ under the condition $s \mid k$.
• To determine the spectrum of $c$-colourings of type $s$ for which each vertex is incident to exactly $s$ colours.
• To establish the lower and upper bounds for the $s$-chromatic index in such systems.
• To construct explicit equitable colourings for a wide range of $c$ values using combinatorial block decomposition techniques.
• To prove that the minimum $s$-chromatic index is exactly $s$ when $s \mid k$.

Proposed method

• Constructs a $4$-cycle system $\Sigma = (X, \mathcal{B})$ of order $v = 1 + 8k$ using disjoint sets $A_i$ and a central vertex $\infty$, with blocks formed from $\mathcal{B}_i$ and cross-sets $[A_p, A_q]$.
• Defines a colouring $f$ by assigning colours based on modular arithmetic: $f([A_p, A_q]) = i$ if $p + q \equiv i \pmod{s+1}$ for $c = s+1$.
• Extends colourings by assigning new colours to selected $[A_p, A_q]$ blocks in a way that avoids colour conflicts and maintains equitable distribution.
• For higher $c$, decomposes the complete graph $K_{2s}$ minus a $1$-factor into $3$- and $4$-cycles, assigning colours via these cycle structures.
• Uses the equation $2s^2 - 2s = 4t \cdot 3 + (\frac{s^2 - s}{2} - 3t) \cdot 4$ to balance colour class sizes in the final construction.
• Applies known results on $1$-factorization and cycle decomposition (from [7, Theorem 2.4]) to partition $K_{2s} - I$ into $3$- and $4$-cycles for colour assignment.

Experimental results

Research questions

• RQ1 For which values of $c$ does an equitable $c$-colouring of type $s$ exist in a $4$-cycle system of order $v = 1 + 8k$ when $s \mid k$?
• RQ2 What is the minimum number of colours required for an equitable block colouring of type $s$?
• RQ3 What is the maximum number of colours possible in such a colouring, and how can it be bounded?
• RQ4 Can equitable colourings be systematically constructed for $c \in \{s, s+1, \dots, \lfloor (2s^2 + s)/3 \rfloor\}$?
• RQ5 How does the structure of the $4$-cycle system and the distribution of blocks around each vertex affect equitable colouring feasibility?

Key findings

• When $s \mid k$, the $s$-chromatic index $\chi'_s(v)$ is exactly $s$, meaning $s$ colours are both necessary and sufficient for an equitable $c$-colouring of type $s$.
• For $c \in \{s, s+1, \dots, \lfloor (2s^2 + s)/3 \rfloor\}$, equitable $c$-colourings of type $s$ exist in $4$-cycle systems of order $v = 1 + 8k$.
• The construction for $c = s+1$ uses $s$ disjoint $4$-cycle systems of order $1 + 8h$ and $s+1$-colouring via modular arithmetic on cross-set blocks.
• For $c > s+1$, the method extends colourings by assigning new colours to $c - s - 1$ selected cross-set blocks $[A_p, A_q]$ without violating equitability.
• For $c \geq \frac{s^2 + s}{2} + 1$, the construction uses a decomposition of $K_{2s} - I$ into $3$- and $4$-cycles to assign colours to $[A_p, A_q]$ blocks, ensuring balanced class sizes.
• The upper bound on the $s$-chromatic index is $\overline{\chi}'_s(v) \leq \frac{s^2 v}{v + s - 1}$, derived from counting vertex incidences and block colour class sizes.